Harmonic functions of polynomial growth on complete manifolds. II (Q1907745)

A complete noncompact Riemannian manifold \(M\) of dimension \(m\) has the strong Liouville property if the dimension of the space \({\mathcal H}_d (M)\) of harmonic functions of polynomial growth of order \(\leq d\) is finite for all \(d\). The author shows that if \(M\) is a Hadamard manifold with volume growth \(V(B (x_0, r))\leq \text{const}\cdot r^m\) and with sectional curvature \(K\geq- \text{const}/ r^2\), then \(M\) has the strong Liouville property. He also shows a corresponding result in the case of nonnegative curvature. In the latter case, stronger results have been announced by T. Colding and W. Minicozzi.